generalization and resolution of the homogeneous …€¦ · based on the theory of factorization...

Advances in Differential Equations and Control Processes © 2016 Pushpa Publishing House, Allahabad, India Published: November 2016 http://dx.doi.org/10.17654/DE017040265 Volume 17, Number 4, 2016, Pages 265-283 ISSN: 0974-3243

Received: May 29, 2016; Accepted: August 14, 2016

2010 Mathematics Subject Classification: 34A05, 34C20, 35K57, 92D25.

Keywords and phrases: homogeneous reaction-diffusion equations, method of factorization of ordinary differential operators, population dynamics. ∗Corresponding author

Communicated by Yansheng Liu

GENERALIZATION AND RESOLUTION OF THE HOMOGENEOUS REACTION-DIFFUSION

EQUATIONS BY THE METHOD OF FACTORIZATION OF ORDINARY DIFFERENTIAL OPERATORS

S. Moumouni1,*, V. M. S. Soumanou2, S. Massou2, A. L. Essoun2 and M. Tchoffo3

1Higher Teachers’ Training School of Natitingou University of Parakou 06 BP 1716 Cotonou (République du Bénin) Benin e-mail: [email protected]

2Department of Physics Faculty of Science and Technology University of Abomey-Calavi Benin

3Department of Physics University of Dschang Cameroon

S. Moumouni et al. 266

Abstract

The objective of this article is the generalization of the homogeneous reaction-diffusion equations of the monostable or bistable types. From the non-linear second order differential equation of the front profile (solution of the homogeneous reaction-diffusion equations), a method of generalization of the homogeneous reaction-diffusion equations, based on the theory of factorization of ordinary differential operators [9], was proposed. This method enables us to build two generalized homogeneous reaction-diffusion equations: the first with one species and the second with two species.

1. Introduction

Equations of models of diffusion intervene in varied fields like combustion [1], chemistry [2], biology or ecology [3]. In population dynamics [4], these equations model the evolution of species which interact between themselves and move in an environment. When we consider an isolated species of density ( )xtu , at time t, with the position x and which

moves in the direction ( ),Ox its equation of diffusion can be written in the

following general form:

( ) ( )( ) ( )( ) .,0,,,,, R∈>+=∂

∂ xttxuxftxuDttxu

x (1.1)

In this expression, ( )uxf , represents the reaction term which defines

interactions between the species and its environment. This term is a function of the density ( )txu , of species and their position x. The movement of the

species is described by the dispersion operator .xD According to the mode of

displacement of the species, this operator is local or non-local. Thus, we distinguish two types of non-linear equations of diffusion: reaction-diffusion equations and integro-differential equations.

In the case of the reaction-diffusion equations, the dispersion operator

xD is local because it is supposed that the species at the time t and at

position x diffuse only towards its immediate neighbour. Thus, this operator

Generalization and Resolution of the Homogeneous … 267

is an elliptic differential operator of the second order. The first models of reaction-diffusion introduced by [5, 6] in population genetics are written in the form:

( ) ( ) ( )( ) .,0,,,,2

2R∈>+

∂

∂=∂

∂ xttxufx

txut

txu (1.2)

These reaction-diffusion equations are non-linear parabolic partial derivative equations. In these equations, the non-linearity shown in the reaction term f does not depend on the position x in space. In the case where the environment of the species is homogeneous, we consider the regular reaction term f in the interval [ ]1,0 and null at the ends ( ) ( )( ).010 == ff

This approach of reaction-diffusion becomes inaccurate when we study species whose individuals can carry out long distance displacements. These types of dispersions at long distances intervene in phenomena like the pollination of certain plants or the propagation of epidemics. These dispersions are often due to external vectors (human, animal, ...) which foster the transport of species far from their place of origin. A manner of taking into account these types of dispersions at long distances is to use integro-differential equations [7, 8] in the form:

( ) ( ) ( ) ( ) ( )( )∫∞+

∞−+−−=

∂∂ ,,,,, txuftxudytyuyxJt

txu

0>t and .R∈x (1.3)

In these equations, J is a density of probability also called core of dispersion. Thus, the dispersion operator xD is described by an integral operator of the

form .uuJ − The study of these types of equations is beyond the scope of

this work.

The objective of this article is to generalize and solve the homogeneous equations of reaction-diffusion (1.2) by using the method of factorization of differential operators suggested in [9]. In Section 2, we describe the various types of homogeneous reaction-diffusion equations. In Section 3, we present


the methodology of resolution of the reaction-diffusion equations by the factorization method of operators. Sections 4 and 5 are dedicated to the results of generalizations of the homogeneous reaction-diffusion equations, with one species and two species, respectively. Finally, in Section 6, we report the conclusion of work.

2. Various Types of Homogeneous Reaction-diffusion Equations

Generally, the reaction term f in equation (1.2), used in population dynamics is monostable or bistable.

2.1. Monostable reaction term

The reaction term f is of the monostable type if:

( ) ( ) 01,00 <′>′ ff and 0>f on ] [.1,0 (2.1)

The current traditional example of monostable term is:

( ) ( ) ( )uauuuf 11 +−= with .0≥a (2.2)

If ,1≤a then the line of equation ( )ufy 0′= is above the first bisectrix

which corresponds to a reaction term of type KPP (for Kolmogorov, Petrovsky and Piskunov [6]). If on the other hand, ,1 a> then the line

of equation ( )ufy 0′= is below the first bisectrix, which in ecology,

corresponds to a weak Allee effect [10].

2.2. Bistable reaction term

The reaction term f is of the bistable type if:

( )∫ >1

00dssf with ( ) ,00 <′f and ( ) .01 <′f (2.3)

The traditional example of bistable term the most common is the cubic function:

( ) ( ) ( )uuuuf ρ−−= 1 with ] [.21,0∈ρ (2.4)

Thus, 0<f on ] [ρ,0 and 0>f on ] [.1,ρ


3. Resolution of the Homogeneous Reaction-diffusion Equations by the Method of Factorization of Ordinary Differential Operators

Several authors [6, 11-13] showed that the reaction-diffusion models of the monostable or bistable type have solutions of the fronts type, in uniform translation connecting the state of balance in ∞− to the state of balance in

,∞+ which makes it possible for these models to describe the invasion at

speed and constant profile of a virgin space by a population. These solutions are ( ) ( ),, ctxUtxu −= where c is the speed of the front and U is the front

profile connecting the two states of balance of the equation. The front profile U of equation (1.2) satisfies the following non-linear elliptic equation:

( ) ( ) ( )( ) ,02

2=++ zUfdz

zdUcdz

zUd where ( ) R∈−= ctxz

and .10 << U (3.1)

According to [9], equation (3.1) can be factorized in form:

( )( ) ( )( ) ( ) .021 =

Ψ−

Ψ− zUzUdz

dzUdzd (3.2)

The expansion of (3.2) leads to the following equation:

( ) ( ) ( ) .0212

212

2=ΨΨ+

Ψ+Ψ+Ψ− zUdz

zdUUdUd

dzzUd (3.3)

Thus, the particular solution of equation (3.1) is one of the solutions of the equations below:

( )( ) ( ) ,01 =

Ψ− zUzUdz

d (3.4a)

( )( ) ( ) 02 =

Ψ− zUzUdz

d (3.4b)


provided that 1Ψ and ,2Ψ solutions of system (3.5) below exist:

( )

−=Ψ

+Ψ+Ψ

=ΨΨ

.

,

221

21

cUdUdUfU

(3.5)

In the continuation of this work, we will use this approach to generalize and solve the homogeneous reaction-diffusion of equations for one species and two species, respectively.

4. Generalization of the Homogeneous Reaction-diffusion Equations to One Species

4.1. First stage of generalization

In the expression of the reaction term f of the traditional reaction-

diffusion models (monostable or bistable), the growth rate ( )uuf comprises

two terms: the first term is in favor of the growth of the species; and the second term is against its growth. This second term still called function of braking is often generalized by several authors including [14] in the form:

( ) .,1 +∈− Rnun

In this work, we adopt this generalized form of the function of braking, which leads us to write the reaction term in equation (1.2) in the form:

( ) ( ) ( ) .1 uuguuf n−= (4.1)

With this stage, our objective is to determine the expression of the function so that the reaction-diffusion equation (1.2) with the reaction term (4.1) can admit a particular solution with the step described in Section 3. Thus, by taking into account the system (3.5), we can choose the functions

1Ψ and 2Ψ such as:

( ) ( )nUkU −=Ψ 11 and ( ) .,12

∗∈=Ψ RkUgk (4.2)


And g must satisfy the following differential equation:

( ) ( ) ( ) .012 =+−++′ kcUkUgUgU n (4.3)

By solving this equation and taking into account that ( ) ( ) ,010 == ff we

obtain:

( ) ( ).1

2ckkUn

kUg n +−+

= (4.4)

Thus, we obtain a generalized expression of the reaction term in the following form:

( ) ( ) ( ) ,1 uauuuf nn ρ−−= where ∗+∈ Ra and .∗∈ρ R (4.5)

By considering the expression (4.5) for the reaction term, the functions

1Ψ and 2Ψ are as follows:

( ) ( )nUkU −=Ψ 11 and ( ) .,12

∗∈ρ−=Ψ RkaUkn (4.6)

Thus, we use these expressions to solve equations (3.4a) and (3.4b). Nevertheless, only the solution obtained with equation (3.4b) is not valid and is as follows:

( ) ,1exp

exp

−

ρ−

ρ−ρ

=zk

nAa

zknA

zU n where ( )12 += nak and ( ).kck +=ρ (4.7)

In summary, the homogeneous reaction-diffusion equation that we generalized is structured as follows:

( ) ( ) ( ( )) ( ( ) ) ( )txutxautxux

txut

txu nn ,,,1,,2

2ρ−−+

∂

∂=∂

∂

,,0 R∈> xt where ∗+∈ Ra and .∗∈ρ R (4.8)

If moreover, at initial time, ( ) ( ),0, xxu µ= the particular solution

of equation (4.8) obtained by the method of factorization of ordinary


differential operators [9] is as follows:

( ) ( ) ( )( ) ( )( )

,1exp

exp,−ωµ+ρ

ωρµ=

txatxtxu

nn

nn

n where ( )( )( ) .1

1+

+−ρρ=ω nanan

n (4.9)

4.2. Second stage of generalization

In the first stage, we carried out the generalization of the reaction term which represents the only term of non-linearity in the traditional reaction-diffusion equation. In the second stage, we add a second term of non-linearity and rewrite equation (4.8) in this new form:

( )( ) ( ) ( ) ( ( )) ( ( ) ) ( ),,,,1,,, 2

2txutxautxu

xtxu

ttxutxuh nn ρ−−+

∂

∂=∂

∂

,,0 R∈> xt where ∗+∈ Ra and .∗∈ρ R (4.10)

By applying the step described in Section 3, the objective at this stage is to determine the expression of the function h. The front profile U of equation (4.10) satisfies the following non-linear elliptic equation:

( ) ( )( ) ( ) ( ( )) ( ( ) ) ( ) ,012

2=ρ−−++ zUzaUzUdz

zdUczUhdz

zUd nn

where ( ).ctxz −= (4.11)

By taking into account the system (3.5), we obtain the following system:

( ) ( )

( )

( )

∈

−=Ψ+Ψ+Ψ

ρ−=Ψ

−=Ψ

∗.

,

,1,1

221

2

1

Rk

cUhUdUd

aUk

UkU

n

n

(4.12)

The resolution of the last equation of this system gives:

( ) ( ) ,122nUkc

nakkc

kUh +−+−ρ= (4.13)


we can write the function in the form:

( ) ,nuuh β+α= where ∗+∈α R and .R∈β (4.14)

Thus, the second reaction-diffusion equation that we have just generalized is as follows:

( ( )) ( ) ( ) ( ( )) ( ( ) ) ( ),,,,1,,, 2

2txutxautxu

xtxu

ttxutxu nnn ρ−−+

∂∂=

∂∂β+α

,,0 R∈> xt where ( ) ,, 2∗+∈α Ra ∗∈ρ R and .R∈β (4.15)

By supposing that ( ) ( ),0, xxu µ= the particular solution of equation

(4.15) obtained by the method of factorization of ordinary differential operators [9] is in the form:

( ) ( ) ( )( ) ( )( )

,1exp

exp,−ωµ+ρ

ωρµ=

txatxtxu

nn

nn

n where ( )( )( ) .1

1+α+βρ+−ρρ=ω na

nann (4.16)

4.3. Analysis of particular solutions of the generalized homogeneous reaction-diffusion equations

All our analysis will relate to equation (4.8) which corresponds to equation (4.15) when 1=α and .0=β With equation (4.8), we obtain

successively from the reaction term:

( ) ( ) ( ) ( ) ,121 2 ρ−+−ρ++=′ nn unauanuf (4.17)

( ) ρ−=′ 0f and ( ) ( ),1 anf −ρ=′ (4.18)

( ) ( )( )( ) ( )∫ ++

ρ+−=1

0.212

1nn

nandssf (4.19)

4.3.1. Monostable reaction-diffusion model

The reaction-diffusion model is of the monostable type if the reaction term f checks: ( ) ( ) 01,00 <′>′ ff and 0>f on ] [.1,0


Thus, for the model (4.8), it is important that 0<ρ and 0>a

(Figure 1). Under these conditions, 0,0 >∀>ω nn then, this gives rise to a

process of growth of the species as illustrated in Figure 2.

Figure 1. Shape of the reaction term (monostable) of the reaction-diffusion model (4.8), for various values of the parameters. Figure (a) corresponds to the KPP type [6] and Figure (b) corresponds to a weak Allee effect [10].

Figure 2. Shape of the solution (4.9): case of the monostable reaction term. It is about a process of growth of the species because .0,0 >∀>ω nn Here

( ) ( ) ( ) .2exp0, 2 π−=µ= xxxu (a) and (b) are drawn for the same values

of parameters, but we swapped the directions of the (Ox) and (Ot) axes with one another.


4.3.2. Bistable reaction-diffusion model

The reaction-diffusion model is of the bistable type if the reaction term

f checks: ( ) ( ) ( )∫ <′<′>10

.01,00,0 ffdssf Thus, in equation (4.8), it is

necessary to have: 0>ρ and ( )ρ+> 1na (Figure 3).

Figure 3. Shape of the reaction term (bistable) of the reaction-diffusion model (4.8), for various values of the parameters. Figure (a) corresponds to the case of the variation of the parameter a and Figure (b) corresponds to the case of the variation of the parameter ρ .

Under these conditions, we attain two types of processes:

- Growth of the species when 0>ωn if ( )an 1+>ρ as illustrated in

Figure 4.

- Decrease of the species when 0<ωn if ( )an 1+<ρ as illustrated in

Figure 5.


Figure 4. Shape of the solution (4.9): case of the bistable reaction term. The

process is a growth when ( ) .1 an +>ρ Here ( ) ( ) ( ) .2exp0, 2 π−=µ= xxxu

(a) and (b) are drawn for the same values of parameters, but we swapped the directions of the (Ox) and (Ot) axes with one another.

Figure 5. Shape of the solution (4.9): case of the bistable reaction term. The

process is a decline when ( ) .1 an +<ρ Here ( ) ( ) ( ) .2exp0, 2 π−=µ= xxx

(a) and (b) are drawn for the same values of parameters, but we swapped the directions of the (Ox) and (Ot) axes with one another.

For example, when the process of diffusion is decreasing, one is

interested in the half-life time noted ∗T such as: ( ) ( ) .20,, xuTxu =∗ For


equation (4.8) whose solution is (4.9), we obtain:

( ) ( )( )

.2

ln1

µ−ρµ−ρ

ω=∗

xaxaxT nn

n

n (4.20)

Figure 6(a) describes the variation of half-life time at a particular point with the parameter. On the other hand, Figure 6(b) describes the variation of half-life time with x.

Figure 6. Shape of the half-life time (4.20). Figure (a) describes its variation at one particular point 00 =x with the parameter n. On the other

hand, Figure (b) describes its variation with x. Here ( ) ( ) =µ= xxu 0,

( ) .2exp 2 π−x

5. Generalization of the Homogeneous Reaction-diffusion Equations to Two Species

The objective of this section is to propose a generalization of the homogeneous reaction-diffusion equation to two species and to solve this equation by using the method of factorization of ordinary differential operators [9]. In population dynamics, there are various types of interactions between two species. Three most significant and documented interactions are: the predation, the competition and the mutualism. In case of our study, we choose to write a general system, without specifying the type of


interaction:

( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )

∈>

+∂

∂=∂

∂

+∂

∂=∂

∂

,,0

,,,,,,,,,,

,,,,,,,,,,

22

22

12

21

Rxt

txvtxvtxufx

txvt

txvtxvtxug

txutxvtxufx

txut

txutxvtxug

(5.1)

where ( )txu , and ( )txv , are the densities of each species at time t and

position x. As in the case with a species, we are interested in the front profiles of the same front speed c such as: ( ) ( )ctxUtxu −=, and

( ) ( )., ctxVtxv −= We also place ourselves under the conditions of Cauchy

such as: ( ) ( )xxu µ=0, and ( ) ( ).0, xxv ϑ= The differential connection of

the front profiles to the system (5.1) is:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

=++

=++

,0,,

,0,,

222

2

112

2

zVVUfdzzdVVUcg

dzzVd

zUVUfdzzdUVUcg

dzzUd

where ( ).ctxz −= (5.2)

By applying the approach of factorization of ordinary differential operators [5] to the system (5.2), we have:

( ) ( ) ( )

( ) ( ) ( )

=

Ψ−

Ψ−

=

Ψ−

Ψ−

,0,,

,0,,

2221

1211

zUVUdzdVUdz

d

zUVUdzdVUdz

d

(5.3)

we obtain, after expansion of (5.3), the following system:


( ) ( ) ( )

( ) ( ) ( )

=ΨΨ+

∂Ψ∂

+Ψ+Ψ−

=ΨΨ+

∂Ψ∂

+Ψ+Ψ−

.0

,0

222122

22212

2

121112

12112

2

zVdzzdVVVdz

zVd

zUdzzdUUUdz

zUd

(5.4)

By comparing the systems (5.2) and (5.4), we successively obtain:

( ) ( ) ( )

( )

−=∂Ψ∂

+Ψ+Ψ

=ΨΨ

,,

,,,,

112

1211

11211

VUcgUU

VUfVUVU (5.5a)

( ) ( ) ( )

( )

−=∂Ψ∂

+Ψ+Ψ

=ΨΨ

.,

,,,,

222

2221

22221

VUcgVV

VUfVUVU (5.5b)

The generalization that we propose in this case is similar to that of the process with one species. Thus, we define the functions 1f and 2f as

follows:

[ ] [ ] R→× 1,01,0:1f

( ) ( ) ( ),, 1111 γ−ρ− mn VbUaVU (5.6a)

[ ] [ ] R→× 1,01,0:2f

( ) ( ) ( ),, 2222 γ−ρ− mn VbUaVU (5.6b)

( ) ,, 2∗+∈ Rmn ( ) 4

2121 ,,, ∗∈ Rbbaa and ( ) 42121 ,,, ∗∈γγρρ R are the

parameters of the model. At this stage, the objective of the work is to determine the functions 1g and ,2g and the conditions on the parameters

so that the system (5.2) admits at least a particular solution by the method of factorization of ordinary differential operators [9]. By replacing the expressions (5.6a) and (5.6b), respectively, in the systems (5.5a) and (5.5b) and after resolution, the following expressions for the functions are obtained:


[ ] [ ] R→× 1,01,0:1f

( ) ( ) ( ),, γ−ρ− nn aVaUVU (5.7a)

[ ] [ ] R→× 1,01,0:2f

( ) ( ) ( ),, ρ−γ− nn bVbUVU (5.7b)

[ ] [ ] R→× 1,01,0:1g

( ) ( ),, nn VUaVU +ε−δ (5.7c)

[ ] [ ] R→× 1,01,0:2g

( ) ( )., nn VUbVU +ε−δ (5.7d)

With ,∗+∈ Rn ( ) ,, 2∗+∈ Rba ( ) 3,, ∗∈δγρ R and R∈ε which are the new

parameters of the model. In addition:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

ρ−=Ψγ−=Ψ

ρ−=Ψγ−=Ψ

,,and1,

,,and1,

2222

21

1121

11

nn

nn

bVkVUbUkVU

aUkVUaVkVU

where ( ) ., 221

∗∈ Rkk (5.8)

By introducing the expressions (5.8) into equations (5.4), we obtain the solution of system (5.2). Thus, the system of reaction-diffusion equations to two species that we have just built is as follows:

[ ( )] ( ) ( )

[ ( )] ( ) ( )

∈>

ρ−γ−+∂∂=

∂∂+ε−δ

γ−ρ−+∂∂=

∂∂+ε−δ

.,0

,

,

2

2

2

2

Rxt

vbvbux

vtvvub

uavauxu

tuvua

nnnn

nnnn

(5.9)


And its particular solution is:

( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )

−θϑ+ρ

θρϑ=

−θµ+ρ

θρµ=

,1exp

exp,

,1exp

exp,

txbtxtxv

txatxtxu

nn

nn

n

nn

nn

n

(5.10)

where ( )( )( ) .1

1+δ

+γ+ρρ=θ nnn

n

6. Conclusion

This work has led to the analysis of the homogeneous reaction-diffusion equations whose terms of reactions are of the monostable or bistable type, and where the density of the species is denoted by ( )., txu Several former

works had shown that these reaction-diffusion equations have solutions of front type, in uniform translation connecting the state of balance at ∞− to the

state of balance at .∞+ These solutions are ( ) ( ),, ctxUtxu −= where c is

the speed of the front and U is the front profile. The front profile U checks a non-linear differential equation of the second order. In addition, the method of factorization of ordinary differential operators, recently, published by [9], opened a new step for the resolution of the differential equations. By applying this step to the differential equation of the front profile U, we proposed a method of generalization of the traditional homogeneous reaction-diffusion equations (monostable and bistable). This method has enabled us to build two generalized homogeneous reaction-diffusion equations: the first with one species and the second with two species. It also made it possible to determine the particular solution of each equation. These generalizations made it possible to increase the number of parameters of the traditional reaction-diffusion equations, thus, offering more flexibility for the adjustment of the solutions ( )txu , on measured data.


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